Quantum Hoare Logic with Classical Variables

Hoare logic provides a syntax-oriented method to reason about program correctness and has been proven effective in the verification of classical and probabilistic programs. Existing proposals for quantum Hoare logic either lack completeness or support only quantum variables, thus limiting their capability in practical use. In this article, we propose a quantum Hoare logic for a simple while language that involves both classical and quantum variables. Its soundness and relative completeness are proven for both partial and total correctness of quantum programs written in the language. Remarkably, with novel definitions of classical-quantum states and corresponding assertions, the logic system is quite simple and similar to the traditional Hoare logic for classical programs. Furthermore, to simplify reasoning in real applications, auxiliary proof rules are provided that support standard logical operation in the classical part of assertions and super-operator application in the quantum part. Finally, a series of practical quantum algorithms, in particular the whole algorithm of Shor’s factorisation, are formally verified to show the effectiveness of the logic.

Gravitational shock waves and scattering amplitudes

We study gravitational shock waves using scattering amplitude techniques. After first reviewing the derivation in General Relativity as an ultrarelativistic boost of a Schwarzschild solution, we provide an alternative derivation by exploiting a novel relation between scattering amplitudes and solutions to Einstein field equations. We prove that gravitational shock waves arise from the classical part of a three point function with two massless scalars and a graviton. The region where radiation is localized has a distributional profile and it is now recovered in a natural way, thus bypassing the introduction of singular coordinate transformations as used in General Relativity. The computation is easily generalized to arbitrary dimensions and we show how the exactness of the classical solution follows from the absence of classical contributions at higher loops. A classical double copy between gravitational and electromagnetic shock waves is also provided and for a spinning source, using the exponential form of three point amplitudes, we infer a remarkable relation between gravitational shock waves and spinning ones, also known as gyratons. Using this property, we infer a family of exact solutions describing gravitational shock waves with spin. We then compute the phase shift of a particle in a background of shock waves finding agreement with an earlier computation by Amati, Ciafaloni and Veneziano for particles in the high energy limit. Applied to a gyraton, it provides a result for the scattering angle to all orders in spin.

Speed improvement of the quantum factorization algorithm of P. Shor by upgrade its classical part

This report discusses Shor’s quantum factorization algorithm and ρ–Pollard’s factorization algorithm. Shor’s quantum factorization algorithm consists of classical and quantum parts. In the classical part, it is proposed to use Euclidean algorithm, to find the greatest common divisor (GCD), but now exist large number of modern algorithms for finding GCD. Results of calculations of 8 algorithms were considered, among which algorithm with lowest execution rate of task was identified, which allowed the quantum algorithm as whole to work faster, which in turn provides greater potential for practical application of Shor’s quantum algorithm. Standard quantum Shor’s algorithm was upgraded by replacing the binary algorithm with iterative shift algorithm, canceling random number generation operation, using additive chain algorithm for raising to power. Both Shor’s algorithms (standard and upgraded) are distinguished by their high performance, which proves much faster and insignificant increase in time in implementation of data processing. In addition, it was possible to modernize Shor’s quantum algorithm in such way that its efficiency turned out to be higher than standard algorithm because classical part received an improvement, which allows an increase in speed by 12%.

Occupational pension schemes for part-time workers: Equality in the eye of the beholder?

Although much attention is given to the newer forms of atypical work, a large percentage of employees still have a ‘classical’ part-time employment contract. 1 Despite long-standing legal protection against discrimination, these part-time workers risk receiving less favourable treatment. This article discusses the principle of non-discrimination in the field of occupational pensions from the perspective of part-time employment. It aims to show the adverse impact that seemingly neutral measures can have on the occupational pensions of part-time workers. By way of illustration, two cases of the Court of Justice of the European Union: the Schönheit and Becker case (length of service) and the Kleinsteuber case (split pension formula) are examined in detail. It is further argued that, in Kleinsteuber, the CJEU appears to have broadened the possibilities of justifying differential treatment. This tendency could threaten the efficiency of the principle of equal pay and non-discrimination for part-time workers as regards occupational pensions. Finally, the impact of these judgments on Belgian and Dutch occupational pension schemes is examined.

The 2014 International Planning Competition: Progress and Trends

We review the 2014 International Planning Competition (IPC-2014), the eighth in a series of competitions starting in 1998. IPC-2014 was held in three separate parts to assess state-of-the-art in three prominent areas of planning research: the deterministic (classical) part (IPCD), the learning part (IPCL), and the probabilistic part (IPPC). Each part evaluated planning systems in ways that pushed the edge of existing planner performance by introducing new challenges, novel tasks, or both. The competition surpassed again the number of competitors than its predecessor, highlighting the competition’s central role in shaping the landscape of ongoing developments in evaluating planning systems.

Cosmological constant and Euclidean space from nonperturbative quantum torsion

Heisenberg's nonperturbative quantization technique is applied to the nonperturbative quantization of gravity. An infinite set of equations for all Green's functions is obtained. An approximation is considered where: (a) the metric remains as a classical field; (b) the affine connection can be decomposed into classical and quantum parts; (c) the classical part of the affine connection is the Christoffel symbols; (d) the quantum part is the torsion. Using a scalar and vector fields approximation it is shown that nonperturbative quantum effects give rise to a cosmological constant and an Euclidean solution.

High-Strength Steel Door-Beam Design and Digital Simulation

Due to the yield strength σs, tensile strength σb of high-strength steel is much higher than original one, it is applied in making auto’s structure parts to improve the security of passengers and reduce the weight of auto. To make the production of the cold-stamp parts easier, the paper analyzed the structure of a classical part, door-beam, and found that the surface of the part should be G1 continuity at least, rather than G0 continuity. The G1 continuity part has more uniform thickness, less inner strain, and more safety from cracking.